Certainly! Let's go through each part of the question step by step.
Given:
[tex]\[ f(x) = x - 5 \][/tex]
[tex]\[ g(x) = x^2 - 1 \][/tex]
We'll perform the operations as requested:
### (a) [tex]\( f + g \)[/tex]
To find [tex]\( f + g \)[/tex], we simply add [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) + g(x) = (x - 5) + (x^2 - 1) \][/tex]
[tex]\[ = x^2 + x - 6 \][/tex]
Let's evaluate this equation at [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) + g(2) = (2 - 5) + (2^2 - 1) \][/tex]
[tex]\[ = -3 + 3 \][/tex]
[tex]\[ = 0 \][/tex]
Thus, the result is [tex]\( 0 \)[/tex].
### (b) [tex]\( f \cdot g \)[/tex]
To find [tex]\( f \cdot g \)[/tex], we multiply [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) \cdot g(x) = (x - 5) \cdot (x^2 - 1) \][/tex]
Let's evaluate this equation at [tex]\( x=2 \)[/tex]:
[tex]\[ f(2) \cdot g(2) = (2 - 5) \cdot (2^2 - 1) \][/tex]
[tex]\[ = -3 \cdot 3 \][/tex]
[tex]\[ = -9 \][/tex]
Thus, the result is [tex]\( -9 \)[/tex].
### (c) Repeat of [tex]\( f \cdot g \)[/tex]
Since part (c) is identical to part (b), the result remains:
[tex]\[ -9 \][/tex]
### (d) [tex]\( f / g \)[/tex]
To find [tex]\( f / g \)[/tex], we divide [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\[ \frac{f(x)}{g(x)} = \frac{x - 5}{x^2 - 1} \][/tex]
Let's evaluate this equation at [tex]\( x=2 \)[/tex]:
[tex]\[ \frac{f(2)}{g(2)} = \frac{2 - 5}{2^2 - 1} \][/tex]
[tex]\[ = \frac{-3}{3} \][/tex]
[tex]\[ = -1 \][/tex]
Thus, the result is [tex]\( -1.0 \)[/tex].
### (e) [tex]\( g / f \)[/tex]
To find [tex]\( g / f \)[/tex], we divide [tex]\( g(x) \)[/tex] by [tex]\( f(x) \)[/tex]:
[tex]\[ \frac{g(x)}{f(x)} = \frac{x^2 - 1}{x - 5} \][/tex]
Let's evaluate this equation at [tex]\( x=2 \)[/tex]:
[tex]\[ \frac{g(2)}{f(2)} = \frac{2^2 - 1}{2 - 5} \][/tex]
[tex]\[ = \frac{3}{-3} \][/tex]
[tex]\[ = -1 \][/tex]
Thus, the result is [tex]\( -1.0 \)[/tex].
### Summary of results:
(a) [tex]\( 0 \)[/tex]
(b) [tex]\( -9 \)[/tex]
(c) [tex]\( -9 \)[/tex]
(d) [tex]\( -1.0 \)[/tex]
(e) [tex]\( -1.0 \)[/tex]
